Solutions to general elliptic equations on nearly geodesically convex domains with many critical points

TL;DR

This paper establishes existence results for solutions to general elliptic equations on nearly geodesically convex domains with many critical points in Riemannian manifolds, expanding understanding of nonlinear PDEs in geometric contexts.

Contribution

It introduces new existence theorems for elliptic equations on complex domains with multiple critical points in Riemannian manifolds.

Findings

Existence of positive solutions on domains with many critical points

Construction of solutions on nearly geodesically convex domains

Results applicable to nonlinear elliptic equations in geometric settings

Abstract

Consider a complete $d$-dimensional Riemannian manifold $(\mathcal M,g)$, a point $p\in\mathcal M$ and a nonlinearity $f(q,u)$ with $f(p,0)>0$. We prove that for any odd integer $N\ge3$, there exists a sequence of smooth domains $\Omega_k\subset\mathcal M$ containing $p$ and corresponding positive solutions $u_k:\Omega_k\to\R^+$ to the Dirichlet boundary problem